Geometric representation of real numbers. Geometric representation and trigonometric form of complex numbers Geometric representation of a set of real numbers
There are the following forms of complex numbers: algebraic(x + iy), trigonometric(r (cos + isin 
)),
indicative(re i 
).
Any complex number z = x + iy can be represented on the XOU plane as a point A (x, y).
The plane on which complex numbers are depicted is called the plane of the complex variable z (we put the symbol z on the plane).
The OX axis is the real axis, i.e. it contains real numbers. ОУ - imaginary axis with imaginary numbers.
x + iy- algebraic notation of a complex number.
Let's derive the trigonometric form of notation for a complex number.
Substitute the obtained values into the initial form:, i.e.
r (cos
+ isin
)
- trigonometric form of notation of a complex number.
The exponential notation of a complex number follows from Euler's formula: 
,then 
z = re i 
- exponential notation of a complex number.
Actions on complex numbers.
1. addition. z 1 + z 2 = (x1 + iy1) + (x2 + iy2) = (x1 + x2) + i (y1 + y2);
2 ... subtraction. z 1 -z 2 = (x1 + iy1) - (x2 + iy2) = (x1-x2) + i (y1-y2);
3. multiplication. z 1 z 2 = (x1 + iy1) * (x2 + iy2) = x1x2 + i (x1y2 + x2y1 + iy1y2) = (x1x2-y1y2) + i (x1y2 + x2y1);
4
... division. z 1 / z 2 = (x1 + iy1) / (x2 + iy2) = [(x1 + iy1) * (x2-iy2)] / [(x2 + iy2) * (x2-iy2)] = 
Two complex numbers that differ only in the sign of the imaginary unit, i.e. z = x + iy (z = x-iy) are called conjugate.
Work.
z1 = r (cos 
+ isin 
); z2 = r (cos 
+ isin 
).
Then the product z1 * z2 of complex numbers is:, i.e. the modulus of the product is equal to the product of the moduli, and the argument of the product is equal to the sum of the arguments of the factors.
;
;
Private.
If complex numbers are in trigonometric form.
If complex numbers are exponential.
Exponentiation.
1. A complex number is given in algebraic form.
z = x + iy, then z n is found from binomial Newton formula:
- the number of combinations of n elements of m each (the number of ways in how many n elements from m can be taken).
; n! = 1 * 2 * ... * n; 0! = 1; 
.
We apply for a complex number.
In the resulting expression, you need to replace the powers of i with their values:
i 0 = 1 Hence, in the general case, we obtain: i 4k = 1
i 1 = i i 4k + 1 = i
i 2 = -1 i 4k + 2 = -1
i 3 = -i i 4k + 3 = -i
Example.
i 31 = i 28 i 3 = -i
i 1063 = i 1062 i = i
2. trigonometric form.
z = r (cos 
+ isin 
), then
- Moivre formula.
Here n can be either “+” or “-” (integer).
3. If a complex number is given in indicative form:
Extracting the root.
Consider the equation: 
.
Its solution will be the nth root of the complex number z: 
.
The nth root of a complex number z has exactly n solutions (values). A root of an effective n-th power has only one solution. In complex - n solutions.
If a complex number is given in trigonometric form:
z = r (cos 
+ isin 
), then the n-th root of z is found by the formula:
, where k = 0.1 ... n-1.
Rows. Number series.
Let the variable a take successively the values a 1, a 2, a 3,…, a n. This renumbered set of numbers is called a sequence. It is endless.
The number series is the expression a 1 + a 2 + a 3 + ... + a n + ... = 
... The numbers a 1, a 2, a 3,…, and n are the members of the series.
For instance.
and 1 is the first member of the series.
and n is the n-th or common term of the series.
A series is considered given if the n-th (common term of the series) is known.
A numerical series has an infinite number of members.
Numerators - arithmetic progression (1,3,5,7…).
the n-th term is found by the formula a n = a 1 + d (n-1); d = a n -a n-1.
Denominator - geometric progression... b n = b 1 q n-1; 
.
Consider the sum of the first n terms of the series and denote it by Sn.
Sn = a1 + a2 +… + a n.
Sn is the n-th partial sum of the series.
Consider the limit: 
S is the sum of the series.
A number of convergent if this limit is finite (a finite limit S exists).
Row divergent if this limit is infinite.
In the future, our task is as follows: to establish which row.
One of the simplest, but often encountered, series is a geometric progression.
, C = const.
The geometric progression isconverging
near, if 
, and divergent if 
.
Also found harmonic series(row 
). This row divergent
.
Geometrically, real numbers, as well as rational numbers, are represented by points on a straight line.
Let l - an arbitrary line, and O - some of its point (Fig. 58). To every positive real number α we put in correspondence the point A, lying to the right of O at a distance of α units of length.
If, for example, α = 2.1356 ..., then
2 < α
< 3
2,1 < α
< 2,2
2,13 < α
< 2,14
and so on. Obviously, point A in this case must be on the straight line l to the right of the points corresponding to the numbers
2; 2,1; 2,13; ... ,
but to the left of the points corresponding to the numbers
3; 2,2; 2,14; ... .
It can be shown that these conditions determine on the line l the only point A, which we consider as a geometric image of a real number α = 2,1356... .
Likewise, for every negative real number β we put in correspondence the point B, which lies to the left of O at a distance of | β | units of length. Finally, we assign the point O to the number "zero".
So, the number 1 will be displayed on the line l point A, located to the right of O at a distance of one unit of length (Fig. 59), the number - √2 - point B, lying to the left of O at a distance of √2 units of length, etc.
Let's show how on a straight line l using a compass and a ruler, you can find the points corresponding to the real numbers √2, √3, √4, √5, etc. To do this, first of all, we will show how you can construct segments, the lengths of which are expressed by these numbers. Let AB be a segment taken as a unit of length (Fig. 60).
At point A, we raise the perpendicular to this segment and put on it the segment AC, equal to the segment AB. Then, applying the Pythagorean theorem to the right-angled triangle ABC, we obtain; ВС = √АВ 2 + АС 2 = √1 + 1 = √2
Consequently, the segment BC has length √2. Now we will restore the perpendicular to the segment BC at point C and select point D on it so that the segment CD is equal to the unit of length AB. Then from the rectangular triangle BCD we find:
ВD = √ВC 2 + СD 2 = √2 + 1 = √3
Therefore, the segment BD has length √3. Continuing the described process further, we could get the segments BE, BF, ..., the lengths of which are expressed by the numbers √4, √5, etc.
Now on the straight l it is easy to find those points that serve as a geometric representation of the numbers √2, √3, √4, √5, etc.
Putting, for example, to the right of the point O the segment BC (Fig. 61), we get the point C, which serves as a geometric image of the number √2. In the same way, laying off the segment BD to the right of the point O, we get the point D ", which is the geometric image of the number √3, and so on.
However, one should not think that with the help of a compass and a ruler on the number line l you can find a point corresponding to any given real number. It has been proved, for example, that, having only a compass and a ruler at our disposal, it is impossible to construct a segment, the length of which is expressed by the number π = 3.14 .... Therefore, on the number line l with the help of such constructions it is impossible to specify a point corresponding to this number. Nevertheless, such a point exists.
So, every real number α can be associated with some well-defined point of the straight line l ... This point will be spaced from the starting point O at a distance of | α | units of length and be located to the right of O if α > 0, and to the left of О, if α < 0. Очевидно, что при этом двум неравным действительным числам будут соответствовать две различные точки прямой l ... Indeed, let the number α corresponds to point A, and the number β - point B. Then, if α > β , then A will be to the right of B (Fig. 62, a); if α < β , then A will lie to the left of B (Fig. 62, b).
Speaking in § 37 about the geometric representation of rational numbers, we posed the question: can any point of the line be regarded as a geometric image of some rational numbers? Then we could not give an answer to this question; now we can answer it quite definitely. There are points on the line that serve as a geometric representation of irrational numbers (for example, √2). Therefore, not every point on the line represents a rational number. But in this case, another question arises: can any point of the number line be considered as a geometric image of some actual numbers? This issue is already being resolved positively.
Indeed, let A be an arbitrary point of the straight line l lying to the right of O (Fig. 63).
The length of the segment OA is expressed by some positive real number α (see § 41). Therefore, point A is the geometric image of the number α ... Similarly, it is established that each point B lying to the left of O can be considered as a geometric image of a negative real number - β , where β is the length of the VO segment. Finally, point O serves as a geometric representation of the number zero. It is clear that two different points of the straight line l cannot be geometrically the same real number.
For the reasons stated above, the straight line on which some point O is indicated as "initial" (for a given unit of length) is called number line.
Conclusion. The set of all real numbers and the set of all points of the number line are in one-to-one correspondence.
This means that each real number corresponds to one, well-defined point of the number line and, conversely, to each point of the number line, with such a correspondence, there corresponds one, well-defined real number.
Complex numbers
Basic concepts
The initial data on the number date back to the Stone Age - paleomelite. These are "one", "little" and "many". They were recorded in the form of notches, nodules, etc. The development of labor processes and the emergence of property forced a person to invent numbers and their names. Natural numbers were the first to appear N received when counting items. Then, along with the need for counting, people had a need to measure lengths, areas, volumes, time and other quantities, where it was necessary to take into account parts of the measure used. This is how fractions arose. The formal substantiation of the concepts of fractional and negative numbers was carried out in the 19th century. Lots of integers Z Are natural numbers, natural numbers with minus and zero signs. Integers and fractional numbers formed a set of rational numbers Q, but it also turned out to be insufficient for the study of continuously changing variables. Genesis again showed the imperfection of mathematics: the impossibility of solving an equation of the form X 2 = 3, in connection with which irrational numbers appeared I. Union of the set of rational numbers Q and irrational numbers I- a set of real (or real) numbers R... As a result, the number line was filled: a point on it corresponded to each real number. But on the set R there is no way to solve an equation of the form X 2 = – a 2. Consequently, the need arose again to expand the concept of number. This is how complex numbers appeared in 1545. Their creator J. Cardano called them "purely negative." The name "imaginary" was introduced in 1637 by the Frenchman R. Descartes, in 1777 Euler proposed to use the first letter of the French number i to denote an imaginary unit. This symbol came into general use thanks to K. Gauss.
During the 17th and 18th centuries, the discussion of the arithmetic nature of imaginations and their geometric interpretation continued. The Dane G. Wessel, the Frenchman J. Argan and the German K. Gauss independently proposed to represent a complex number by a point on the coordinate plane. Later it turned out that it is even more convenient to represent the number not by the point itself, but by a vector going to this point from the origin of coordinates.
Only by the end of the 18th and the beginning of the 19th century did complex numbers take their rightful place in mathematical analysis. Their first use was in the theory of differential equations and in the theory of hydrodynamics.
Definition 1.Complex number is called an expression of the form, where x and y Are real numbers, and i Is an imaginary unit,.
Two complex numbers and are equal if and only if,.
If, then the number is called purely imaginary; if, then the number is a real number, this means that the set R WITH, where WITH- a set of complex numbers.
Conjugated to a complex number is called a complex number.
Geometric representation of complex numbers.
Any complex number can be represented by a dot M(x, y) plane Oxy. A pair of real numbers denotes the coordinates of the radius vector 
, i.e. between the set of vectors on the plane and the set of complex numbers, a one-to-one correspondence can be established:.
Definition 2.The real part X.
Designation: x= Re z(from Latin Realis).
Definition 3.The imaginary part a complex number is called a real number y.
Designation: y= Im z(from Latin Imaginarius).
Re z is plotted on the axis ( Oh), Im z is plotted on the axis ( Oy), then the vector corresponding to the complex number is the radius vector of the point M(x, y), (or M(Re z, Im z)) (Fig. 1).
Definition 4. The plane, the points of which are assigned a set of complex numbers, is called complex plane... The abscissa axis is called real axis since it contains real numbers. The ordinate axis is called imaginary axis, it contains purely imaginary complex numbers. The set of complex numbers is denoted WITH.
Definition 5.Module complex number z = (x, y) is the length of the vector:, i.e. 
.
Definition 6.The argument complex number is the angle between the positive direction of the axis ( Oh) and vector: 
.
The concepts of "set", "element", "belonging of an element to a set" are the primary concepts of mathematics. A bunch of- any collection (collection) of any items .
A is a subset of the set B, if each element of the set A is an element of the set B, i.e. AÌB Û (xÎA Þ xÎB).
Two sets are equal if they consist of the same elements. It's about set-theoretic equality (not to be confused with equality between numbers): A = B Û AÌB Ù BÌA.
Union of two sets consists of elements belonging to at least one of the sets, i.e. хÎАÈВ Û хÎАÚ хÎВ.
Crossing consists of all elements simultaneously belonging to both set A and set B: xÎAÇB Û xÎA Ù xÎB.
Difference consists of all elements A that do not belong to B, i.e. xÎ A \ B Û xÎA ÙxÏB.
Cartesian product C = A´B of sets A and B is called the set of all possible pairs ( x, y), where the first element X each pair belongs to A, and its second element at belongs to V.
The subset F of the Cartesian product A´B is called mapping the set A to the set B if the condition is met: (" XОА) ($! Pair ( xy) ÎF). At the same time, they write: A.
The terms "display" and "function" are synonymous. If ("хÎА) ($! УÎВ): ( x, y) ÎF, then the element atÎ V called way X when displaying F and write it like this: at= F ( X). Element X at the same time is prototype (one of the possible) element y.
Consider the set of rational numbers Q - the set of all integers and the set of all fractions (positive and negative). Each rational number can be represented as a quotient, for example, 1 = 4/3 = 8/6 = 12/9 =…. There are many such ideas, but only one of them is irreducible. .
V Any rational number can be uniquely represented as a fraction p / q, where pÎZ, qÎN, the numbers p, q are coprime.
Properties of the set Q:
1. Closedness with respect to arithmetic operations. The result of addition, subtraction, multiplication, raising to a natural power, division (except division by 0) of rational numbers is a rational number:; ; 
.
2. Ordering: (" x, yÎQ, huy)®( x
Moreover: 1) a> b, b> c Þ a> c; 2)a -b.
3. Density... Between any two rational numbers x, y there is a third rational number (for example, c = ):
("x, yÎQ, x<y) ($ cÎQ): ( X
On the set Q, you can perform 4 arithmetic operations, solve systems of linear equations, but quadratic equations of the form x 2 = a, aÎ N are not always decidable in the set Q.
Theorem. There is no number xÎQ whose square is 2.
g Let there exist a fraction X= p / q, where the numbers p and q are coprime and X 2 = 2. Then (p / q) 2 = 2. Hence,
The right side of (1) is divisible by 2, so p 2 is an even number. Thus, p = 2n (n-integer). Then q must be odd.
Returning to (1), we have 4n 2 = 2q 2. Therefore, q 2 = 2n 2. Similarly, we make sure that q is divisible by 2, i.e. q is an even number. The theorem is proved by contradiction. N
geometric image of rational numbers. Putting off the unit segment from the origin of coordinates 1, 2, 3… times to the right, we get the points of the coordinate line that correspond to natural numbers. Putting aside similarly to the left, we get points corresponding to negative integers. Let's take 1 / q(q = 2,3,4 … ) a part of a unit segment and we will postpone it on both sides of the origin R once. We get points of a straight line corresponding to numbers of the form ± p / q (pÎZ, qÎN). If p, q run through all pairs of coprime numbers, then on the line we have all points corresponding to fractional numbers. In this way, each rational number corresponds, according to the accepted method, to a single point of the coordinate line.
Can a single rational number be indicated for every point? Is the straight line filled entirely with rational numbers?
It turns out that there are points on the coordinate line that do not correspond to any rational numbers. Build an isosceles right-angled triangle on a unit segment. The point N does not correspond to a rational number, since if ON = x- rationally, then x 2 = 2, which cannot be.
There are infinitely many points similar to the point N on the straight line. Take the rational parts of the segment x = ON, those. X... If we put them off to the right, then no rational number will correspond to each of the ends of any of such segments. Assuming that the length of the segment is expressed by a rational number x =, we get that x =- rational. This contradicts what was proved above.
Rational numbers are not enough to associate some rational number with each point of the coordinate line.
Let's build the set of real numbers R across infinite decimal fractions.
According to the "corner" division algorithm, any rational number can be represented as a finite or infinite periodic decimal fraction. When the fraction p / q has no prime factors other than 2 and 5, i.e. q = 2 m × 5 k, then the result will be the final decimal fraction p / q = a 0, a 1 a 2… a n. The rest of the fractions can only have infinite decimal expansions.
Knowing the infinite periodic decimal fraction, you can find the rational number that it represents. But any final decimal fraction can be represented as an infinite decimal fraction in one of the following ways:
a 0, a 1 a 2… a n = a 0, a 1 a 2… a n 000… = a 0, a 1 a 2… (a n -1) 999… (2)
For example, for infinite decimal X= 0, (9) we have 10 X= 9, (9). If we subtract the original number from 10x, we get 9 X= 9 or 1 = 1, (0) = 0, (9).
A one-to-one correspondence is established between the set of all rational numbers and the set of all infinite periodic decimal fractions if the infinite decimal fraction is identified with the digit 9 in the period with the corresponding infinite decimal fraction with the digit 0 in the period according to the rule (2).
Let us agree to use such infinite periodic fractions that do not have the number 9 in the period. If an infinite periodic decimal fraction with the number 9 in a period arises in the process of reasoning, then we will replace it with an infinite decimal fraction with a zero in the period, i.e. instead of 1.999 ... we will take 2,000 ...
Definition of an irrational number. In addition to infinite decimal periodic fractions, there are non-periodic decimal fractions. For example, 0.1010010001 ... or 27.1234567891011 ... (natural numbers are sequentially after the decimal point).
Consider an infinite decimal fraction of the form ± a 0, a 1 a 2 ... a n ... (3)
This fraction is determined by specifying the sign "+" or "-", a non-negative integer a 0 and a sequence of decimal places a 1, a 2, ..., an, ... (the set of decimal places consists of ten numbers: 0, 1, 2, ..., 9).
Any fraction of the form (3) is called real (real) number. If there is a "+" sign in front of the fraction (3), it is usually omitted and written a 0, a 1 a 2 ... a n ... (4)
A number of the form (4) will be called non-negative real number, and in the case when at least one of the numbers a 0, a 1, a 2, ..., a n is different from zero, - positive real number... If in expression (3) the sign "-" is taken, then this is a negative number.
The union of the sets of rational and irrational numbers form the set of real numbers (QÈJ = R). If the infinite decimal fraction (3) is periodic, then this is a rational number, when the fraction is non-periodic, it is irrational.
Two non-negative real numbers a = a 0, a 1 a 2… a n…, b = b 0, b 1 b 2… b n…. are called equal(write a = b), if a n = b n at n = 0,1,2 ... Number a is less than number b(write a<b), if either a 0 or a 0 = b 0 and there is such a number m, what a k = b k (k = 0,1,2, ... m-1), a a m , i.e. a Û (a 0 Ú ($mÎN: a k = b k (k =), a m ). The concept of “ a>b».
To compare arbitrary real numbers, we introduce the concept “ modulus of the number a» . By the modulus of a real number a = ± a 0, a 1 a 2 ... a n ... is called such a non-negative real number represented by the same infinite decimal fraction, but taken with the sign "+", i.e. ½ a½= a 0, a 1 a 2 ... a n ... and 1/2 a½³0. If a - non-negative, b Is a negative number, then it is considered a> b... If both numbers are negative ( a<0, b<0 ), then we will assume that: 1) a = b if ½ a½ = ½ b½; 2) a if ½ a½ > ½ b½.
Properties of the set R:
I. Order properties:
1. For each pair of real numbers a and b there is one and only one relation: a = b, a
2. If a
3. If a , then there is a number c such that a< с .
II. Properties of addition and subtraction actions:
4. a + b = b + a(commutability).
5. (a + b) + c = a + (b + c) (associativity).
6. a + 0 = a.
7. a + (- a) = 0.
8.from a Þ a + c
III. Properties of multiplication and division actions:
9. a × b = b × a .
10. (a × b) × c = a × (b × c).
11. a × 1 = a.
12. a × (1 / a) = 1 (a¹0).
13. (a + b) × c = ac + bc(distribution).
14.if a and c> 0, then a × c
IV. Archimedean property("cÎR) ($ nÎN): (n> c).
Whatever the number cÎR, there is nÎN such that n> c.
V. Continuity property of real numbers. Let two non-empty sets АÌR and BÌR be such that any element aÎА there will be no more ( a£ b) of any element bÎB. Then Dedekind continuity principle asserts the existence of such a number with that for all aОА and bÎB the condition a£ c £ b:
("AÌR, BÌR) :(" aÎA, bÎB ® a£ b) ($ cÎR): (" aÎA, bÎB® a£ c £ b).
We will identify the set R with the set of points of the real line, and call the real numbers points.
CHAPTER 1. Variables and Functions§1.1. Real numbers
The first acquaintance with real numbers occurs in the school mathematics course. Any real number is represented by a finite or infinite decimal fraction.
Real (real) numbers are divided into two classes: the class of rational and the class of irrational numbers. Rational are the numbers that have the form, where m and n Are coprime integers, but 
... (The set of rational numbers is denoted by the letter Q). The rest of the real numbers are called irrational... Rational numbers are represented by a finite or infinite periodic fraction (the same as ordinary fractions), then those and only those real numbers that can be represented by infinite non-periodic fractions will be irrational.
For example, the number 
- rational, and 
, 
, 
etc. - irrational numbers.
Real numbers can also be divided into algebraic - roots of a polynomial with rational coefficients (these include, in particular, all rational numbers - roots of the equation 
) - and on transcendental - all the rest (for example, numbers 
other).
The sets of all natural, whole, real numbers are denoted as follows: NZ, R
(initial letters of the words Naturel, Zahl, Reel).
§1.2. Display of real numbers on the number axis. Intervals
Geometrically (for clarity), real numbers are depicted as dots on an infinite (in both directions) straight line called numerical
axis... For this purpose, a point is taken on the straight line under consideration (the origin is point 0), a positive direction is indicated, depicted by an arrow (usually to the right) and a scale unit is chosen, which is set aside indefinitely in both directions from point 0. This is how integers are depicted. To represent a number with one decimal place, each segment must be divided into ten parts, etc. Thus, each real number will be represented by a point on the number axis. Conversely, to every point 
corresponds to a real number equal to the length of the segment 
and taken with the sign "+" or "-", depending on whether the point lies to the right or to the left of the origin. Thus, a one-to-one correspondence is established between the set of all real numbers and the set of all points of the numerical axis. The terms "real number" and "point of the numerical axis" are used as synonyms.
Symbol 
we will denote both the real number and the point corresponding to it. Positive numbers are located to the right of point 0, negative - to the left. If 
, then on the numerical axis the point 
lies to the left of the point 
... Let the point 
corresponds to a number, then the number is called the coordinate of the point, they write 
; more often the point itself is denoted by the same letter as the number. Point 0 is the origin. The axis is also denoted by the letter 
(Figure 1.1).
Rice. 1.1. Number axis.
The collection of all numbers lying between given numbers and is called an interval or interval; the ends may or may not belong to him. Let's clarify this. Let 
... A collection of numbers that satisfy the condition 
, is called an interval (in the narrow sense) or an open interval, denoted by the symbol 
(Figure 1.2).
Rice. 1.2. Interval
A collection of numbers such that 
is called a closed interval (segment, segment) and is denoted by 
; on the number axis is marked as follows:
Rice. 1.3. Closed interval
It differs from an open gap only in two points (ends) and. But this difference is fundamental, essential, as we will see later, for example, when studying the properties of functions.
Omitting the words "the set of all numbers (points) x such that ", etc., we note further:
and 
, denoted 
and 
half-open, or half-closed, intervals (sometimes: half-intervals);
or 
means: 
or 
and denoted 
or 
;
or 
means 
or 
and denoted 
or 
;
, denoted 
the set of all real numbers. Badges 
symbols of "infinity"; they are called improper or ideal numbers. 
§1.3. The absolute value (or modulus) of a real number
Definition. Absolute value (or modulus) numbers are called this number itself if 
or 
if 
... The absolute value is indicated by the symbol 
... So, 
For instance, 
, 
, 
.
Geometrically means point distance a to the origin. If we have two points and, then the distance between them can be represented as 
(or 
). For instance, 
the distance 
.
Properties of absolute values.
1. It follows from the definition that 
, 
, that is 
.
2. The absolute value of the sum and difference does not exceed the sum of the absolute values: 
.
1) If 
, then 
... 2) If 
, then . ▲
3. 
.
, then by property 2: 
, i.e. 
... Similarly, if we imagine 
, then we come to the inequality 
▲
4. 
- follows from the definition: consider cases 
and 
.
5. 
, provided that 
It also follows from the definition.
6. Inequality 
,
means 
... This inequality is satisfied by the points that lie between 
and 
.
7. Inequality 
tantamount to inequality 
, i.e. ... This is an interval centered at a point of length 
... It is called 
a neighborhood of a point (number). If 
, then the neighborhood is called punctured: this or 
... (Fig. 1.4).
8. 
whence it follows that the inequality 
(
) is equivalent to the inequality 
or 
; and inequality 
defines the set of points for which 
, i.e. these are points that lie outside the segment 
, exactly: 
and 
.
§1.4. Some concepts, designations
Here are some commonly used concepts, designations from set theory, mathematical logic and other branches of modern mathematics.
1 ... Concept multitudes is one of the basic in mathematics, the original, universal - and therefore defies definition. It can only be described (replaced with synonyms): it is a collection, a collection of some objects, things, united by some signs. These objects are called elements sets. Examples: many grains of sand on the shore, stars in the universe, students in the classroom, the roots of an equation, points of a line. Sets whose elements are numbers are called numerical sets... For some standard sets, special designations are introduced, for example, N,Z,R - see § 1.1.
Let A- set and x is its element, then they write: 
; reads " x belongs A» ( 
inclusion sign for elements). If the object x not included in A then write 
; reads: " x do not belong A". For instance, 
N; 8,51
N; but 8.51 
R.
If x is a general designation for the elements of the set A then write 
... If it is possible to write out the designation of all elements, then write 
, 
etc. A set that does not contain any element is called an empty set and is denoted by the symbol ; for example, the set of roots (real) of the equation 
there is an empty one.
The set is called the final if it consists of a finite number of elements. If, for whatever natural number N we take, in the set A there are more than N elements, then A called endless set: there are infinitely many elements in it.
If every element of the set ^ A belongs to the set B, then 
called part or subset of the set B and write 
; reads " A contained in B» ( 
is the inclusion sign for sets). For instance, NZR. If 
, then they say that the sets A and B equal and write 
... Otherwise, write 
... For example, if 
, a 
set of roots of the equation 
, then .
The collection of elements of both sets A and B called amalgamation sets and denoted 
(sometimes 
). A collection of elements belonging to and A and B is called crossing sets and denoted 
... The collection of all elements of the set ^ A that are not contained in B is called difference sets and denoted 
... These operations can be schematically depicted as follows:
If a one-to-one correspondence can be established between the elements of the sets, then they say that these sets are equivalent and write 
... Every multitude A, equivalent to the set of natural numbers N= called counting or countable. In other words, a set is called countable if its elements can be numbered, arranged in an infinite sequence 
, all members of which are different: 
at 
, and it can be written as. Other infinite sets are called uncountable... Countable, except for the set itself N, there will be, for example, the sets 
, Z. It turns out that the sets of all rational and algebraic numbers are countable, and the equivalent sets of all irrational, transcendental, real numbers and points of any interval are uncountable. They say that the latter have the cardinality of the continuum (cardinality is a generalization of the concept of the number (number) of elements for an infinite set).
2
... Let there be two statements, two facts: and 
... Symbol 
means: "if true, then true and" or "from follows", "implies is the root of the equation has the property from English Exist- exist.
Record: 
, or 
, means: there is (at least one) object with the property 
... And the record 
, or 
, means: all have the property. In particular, we can write: 
and .
